The H1 tag is an important element on the list of things to help you to rank.
Whilst not the most important tag it is still a useful way to emphasise the meaning of your content. Both to search engines and human readers.
WordPress for example puts the Title of the individual post into a H1 tag when listing snippets of multiple posts on one page. This would be the logical way to organise the content as it denotes content that is separate to each other, but related to the site topic.
On an individual post however only one H1 tag appears. This is correct as there is only one chunk of distinct information, all other information is of lessor but related importance and is given H2 and H3 tags.
Evidence has shown that the structure of a wordpress blog is exceedingly pleasing to Google. The structure follows logical patterns for the organisation of content. Having multiple H1 tags on one page does not seem to harm the ranking as the content justifies separation from other content on the page. whilst in relation to the site content as a whole.
The contents page of a book would justify chapters being listed in multiple H1 tags. The individual chapter pages would have the chapter title as the H1 tag and nowhere else, if sub headings had to be used it would be the H2 and H3, in order of hierarchy.
Here are some useful links on the matter.
H1, H2, H3 Placement, per W3C standards
Highrankings.com Jan 2006
How important are the “H1″ in SE optimization and why?
Highrankings.com Jan 2007
Importance Of Headings And Sub Headings In Optimizing Pages?
cre8asiteforums.com July 2006
How Important are H1 (Header)Tags
seoroundtable.com August 2005
Search Engine Ranking Factors
SEOmoz.com Date unknown
Blog SEO Tip #7: Heading tags
Stephan Spencer.com Feb 2006
What Attribute Ranks Better Bold or H1 On A Page?
seoroundtable.com March 2005